Lately I’ve been reading the book __Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense__ by Julie McNamara and Meghan Shaughnessy.

I posted the following picture to Twitter while I read during my daughter’s swim class.

My colleague, Hedge, replied about being challenged by a middle school teacher on this very issue.

I let her know I was also challenged about this idea several years ago when I was a digital curriculum developer. The argument I heard back then was that using contexts to validate the correctness of fraction comparisons ran counter to the fact that fractions are numbers. As such, 1/2 is always greater than 1/3 regardless of the context. At the time, I wondered about it, but I still felt that bringing context to bear was important.

Flash forward to now and I have been mulling this idea over all day. I think I may finally understand why we have to be careful what we say about the role of context when comparing fractions. I may be completely off the mark, but I’m going to share my thoughts anyway and let you decide in the comments if you’d like to challenge my thinking or share an alternative point of view.

Let’s start with whole numbers. If I told you to compare 3 and 6, you would probably tell me, “3 is less than 6,” or, “6 is greater than 3.” That is how the numbers 3 and 6 are related.

Now, what if I were to show you these two pictures of 3 and 6: (As illustrated by my daughter’s toys.)

Technically, the 3 dolls are larger and therefore they amount to more *stuff*, but does that really mean 3 is now greater than 6? In the end, the number of dolls my daughter has (3) is less than the number of figurines she has (6). The context doesn’t fundamentally change the relationship between the numbers 3 and 6.

In this case, I don’t even know how I’d justify that she has *more* when referring to the dolls. Sure, they’re bigger, but she may prefer to have more *things* to play with and choose the 6 figurines even though they are less in total size.

Let’s continue by looking at this from a fraction perspective. Now I’m going to take 1/3 of the dolls and 1/2 of the figurines.

In keeping with the idea that context should dictate when one number is greater than another, I should be convinced that 1/3 of the dolls is greater than 1/2 of the figurines because 1 doll is so much larger than the 3 figurines. Oh wait, or is it that I should be thinking that 1/2 of the figurines is greater than 1/3 of the dolls because I end up with 3 figurines which is a greater number of things than 1 doll? It’s not so clear cut, even though I’m trying to let the context dictate how to interpret the fractions.

What it boils down to is that fractions represent a **relationship**. If I think about the relationships each fraction represents, then 1/2 is always greater than 1/3 no matter how I try to spin it. Looking back at my examples, taking 1/2 of the group of figurines means I am taking a greater share of that group (that whole) than when I take 1/3 of the group of dolls (a different whole, but a whole nonetheless). The size of the things in my group (whole) doesn’t matter because the *relationship* represented by 1/2 is greater than the *relationship* represented by 1/3.

Now, does that mean we should ignore contexts altogether? No. There are still rich conversations to be had about who ate more pizza when one person eats half of a small pizza and another person eats a third of a large pizza. Context is still interesting to discuss and helps students use math to interpret the world around them. However, if our goal is to compare fractions, then 1/2 is greater than 1/3 every time.

That’s the argument I came up with today as I tried to understand the criticisms I’ve heard. Now that you’ve read it, what do you think?

mathmindsblogThis conversation is something that always arises in my 5th grade math class every time after I do the lesson in Investigations about different sized wholes. It would even sneak its way into our addition problems. For example, when we would do a number talk like 1/2 + 3/4, they would say, “It would be 1 14/ IF they are the same sized wholes.” I do love when they did that, however it always had me thinking about the same ideas you talked about here but instead of comparison, it was the operations. I loved they way you wrote about comparison Brian, and I don’t know if my thinking is even correct, but it is something I play around with.

I thought about it like this (and talked with the students about it)….when I add naked numbers, without a context, such as 6+3, I am assuming they are talking about the same unit, right? I am assuming we are not talking about 6 feet + 3 inches or 6 Hershey Kisses + 3 Hershey bars….or am I? How does the answer of 9 relate in those situations? 9 units of measure versus “we can’t because they are not the same unit” or 9 pieces of candy versus “we can’t because they are not the same unit. So, not thinking about fractions I think it goes back to what you were talking about….1/2 + 3/4….1/2 of a large pizza + 3/4 of a small pizza = 1 1/4 collection of pizzas versus “we can’t because they not the same sized whole.” Does that hold up there?

Either way, I love that students ask these questions and push us to think about this! To the teacher Hedge is referring to, I would be saddened to think that teachers are not teaching something because it is confusing for students on a the test…these conversations are so valuable and rich in student thinking!

~Kristin

mjrbtonThis whole line of questioning is fascinating to me and makes me think of categorical variables and their role in equivalence relations. The toys were compared using two different categories of attributes/properties: size and amount/number. We often think of equivalence as the realm of numbers and shapes, but the really super fun math conversations (to me at least) are when you get to talk about what exactly it is you’re comparing to determine similarity. So, for example, Hershey kisses and chocolate bars *could* be the same unit if you agree that they are all “candy”. Or, the dolls and the figurines could be categorized as “toys” and you could compare from there. (And, just so you know the depth of my love for comparison, when I was looking at the pictures I saw way more sameness than difference between the dolls/figurines…fun times! 🙂

–malke

bstockusPost authorGreat minds think alike! I was having similar thoughts even as I was writing my post. As long as I could adjust the categories to match, then I could justify my comparison. It’s interesting because this makes sense to students but also adds complexity to their work. They have to learn to negotiate these categories for themselves to understand when and how they can compare or operate on quantities. Sometimes teachers will say things like, “You can’t add apples and pens, that wouldn’t make sense,” but you can if you think of them more generally as objects or things. If the teacher’s logic were universal, then I wouldn’t be able to add boys and girls, but I know in reality I can because they’re all children.

bstockusPost authorI like that your students have had enough experiences thinking about the whole to bring that up when adding fractions. It sounds like they’re proof that getting students to pay attention to the whole doesn’t confuse them to such a degree that they can’t operate on or compare the fractions.

I also like what you said about assuming the whole is the same in a bare naked math problem. It’s a given in that case, but in a real world context we can’t take that for granted. It reminds me of students learning about how to interpret remainders. It can be confusing at first because you have to understand what the question is asking in order to determine what role a remainder might play in your answer. But that confusion isn’t a reason to shy away from the topic. Rather it deepens our understanding of division and how we use it in everyday life.

Simon GreggI think you’re comparison with the number of dolls is really helpful. And I think if you were going to ask this kind of question, it would be best to approach it with the simpler question: Is 6 always bigger than 3?

It’s not about fractions really is it? It’s about that word “of”. Three *of* the dolls. Which is to say,

3 X doll.

I think the question is worth asking, so that students can think about this kind of thing, but I’d start with numbers of the toys first, before fractions.

Simon Gregg*your*

bstockusPost authorGreat advice! I especially like it because this isn’t an issue unique to fractions. As you’re pointing out, it matters anytime we’re comparing quantities of things. I wonder if students have any misunderstandings related to this idea with whole numbers or if it’s something that doesn’t create some possible confusion until fractions. For example, the common example given for paying attention to the whole when comparing fractions is pizzas. I wonder what students would say if you asked them who has more if I have 6 small pizzas and you have 3 large pizzas. In that context with whole numbers is it somehow more obvious that the size of the pizza matters and that 6 is not necessarily greater than 3?

howardat581,2,3,4,5,….. are counting numbers

So there is an implied “thing” of which we have several examples, say apples.

Then 3+6 is a statement about two piles of apples, and counting the whole lot gets you to 9.

3+6=9 is always a shorthand for “three of these together with 6 of these gives you 9 of these”

Context is not just important, it is VITAL.

When we attempt to measure stuff we find that with the chosen unit we may have more than 3 and less than 4 units of stuff, and hence the need for a way to describe “parts of a unit”. This of course leads to fractions, which are essentially measuring numbers. We have NOT extended the number system, we have created a new number system, in which we can talk about “parts”.

So the question “Is a half bigger than a third” is a shorthand for a measurement comparison, such as “with this stick as the unit of length, which of those two sticks is longer?”

I totally agree with Simon Gregg about language, especially “of”. We overlook meaning too easily in math.

tjzagerA problem adapted from Connected Math Project 2 (Lappan et al., 2009):

“Jane and Don’s mathematics classes are selling sub sandwiches as a fund-raiser. Jane’s class has reached 2/3 of their goal and Don’s class has reached 3/4 of their goal. Jane says her class has collected more money than Don’s class.”

“How could Jane be right?”

“How could Jane be wrong?”

I love this problem and think it might throw a monkey wrench in your last couple of paragraphs…

bstockusPost authorThanks for sharing this problem. I love it! 🙂

As I was getting ready to publish my post last night, I got to thinking that the questions we ask is really the interesting part, and the problem you shared is a great example of that. If the question had been, “Which class is closer to its goal?” then 3/4 could be the greater fraction because that class is closer to it goal, whatever it may be. Being 1/4 away from your goal is closer than being 1/3 away from your goal.

However, that may not necessarily be the case in absolute terms because if Don’s class is $100 away from its goal but Jane’s class is $25 away from its goal, then isn’t Jane’s class closer to its goal?

This makes me think of the types of questions we ask to see if students are developing proportional reasoning. For example, Tree A grew so many feet this year and Tree B grew so many feet this year. Which tree grew the most? You can talk about it in terms of how many more feet each tree grew, but that thinking isn’t taking into account growth related to the previous year’s height.

I feel like I’m rambling, but I’m thinking through this as I write. So, if we’re wanting students to be pushed to think proportionally, maybe I would want them to say that Don’t class is closer to its goal regardless of the actual amount of money separating his class from its goal compared to Jane’s class. So in that case, 3/4 would be greater than 2/3 regardless of the amounts of money involved.

That being said, the problem you shared asked a different question about who has raised more money. This shifts the question away from comparing the fractions to one of using the fractions to compare another quantity, the amount of money raised by each class.

tjzagerAlso, thinking about your dolls problem, what happens if you switch them around? Take the larger fraction of the smaller group and vice versa? 1/2 of 3 vs 1/3 of 6?

bstockusPost authorFollowing the logic I was trying to use, does it change anything? If I take 1/2 of the smaller group, I’m still taking more of that group related to its whole. Yes, the specific amount I’m taking (1.5 dolls) is less things than taking 1/3 of the other group (2 figurines) but the relationship involved, taking half of a group, still means I’m taking more of that particular group than taking a third of a group does. I think this gets back to what question is being asked and how that impacts how we need/want to interpret the situation.

howardat58“How could Jane be right?”

“How could Jane be wrong?”

Keeping the feet on the ground, a fraction is a number, but “a fraction of ..” is a quantity and has units. Since the two goals have not been quantified Jane could be right or wrong. Once we know how big (in money terms) the goals are then all will be revealed. So it all goes back to measurement.